Mackey–Arens theorem

The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous dual space. According to Narici (2011), this profound result is central to duality theory; a theory that is "the central part of the modern theory of topological vector spaces."Template:Sfn

Prerequisites

Let Template:Mvar be a vector space and let Template:Mvar be a vector subspace of the algebraic dual of Template:Mvar that separates points on Template:Mvar. If Template:Math is any other locally convex Hausdorff topological vector space topology on Template:Mvar, then we say that Template:Math is compatible with duality between Template:Mvar and Template:Mvar if when Template:Mvar is equipped with Template:Math, then it has Template:Mvar as its continuous dual space. If we give Template:Mvar the weak topology Template:Math then Template:Math is a Hausdorff locally convex topological vector space (TVS) and Template:Math is compatible with duality between Template:Mvar and Template:Mvar (i.e. $X_{\sigma (X,Y)}^{\prime }={\left(X_{\sigma (X,Y)}\right)}^{\prime }=Y$). We can now ask the question: what are all of the locally convex Hausdorff TVS topologies that we can place on Template:Mvar that are compatible with duality between Template:Mvar and Template:Mvar? The answer to this question is called the Mackey–Arens theorem.

See also

• Dual system
• Mackey topology
• Polar topology

Sources

• Template:Rudin Walter Functional Analysis
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels